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Abstract

Polrer Set Models of A-Galeulus: Theories, Expansions, I[sa~raaorphisms
(1) I.S.I., Corso Btalia 40, 56100 Pisa, Italy. Pairtially supported by a grant of the Italian C.N.R. to work at the Laboratory for Computer Science, Massachusetts T~~stitute of
'kchnology, Cam bridge, MA, U.S.A,
This work was supported in pal% by the National Scietice Foundation, Grant No. MCS 8010707, and by a grant to the M.I.T. Laboratory for Compeiter Science by the IBM Corporation,
C-Introduction and summary: n i s paper mainly deals with the models fsr type ficc A-calcu%tasdefined by Plotkin (Plo [1972]) and Engeler (Eng [1979]),
For every nmon empty set A, the model DA is built up In a very natural set theopetic way and provides a code free generalization of early ideas sf Scott, Scott [j1976jv Namely, the notion of application (interpreting forma% applieadon of A-terns) generalizes the classical Payhill-Shepherdson-Rogers definition of application in Pa, introduced to define Enlamesation Operaters (see Ro [9667!,

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For every nmon empty set A, the model DA is built up In a very natural set theopetic way and provides a code free generalization of early ideas sf Scott, Scott [j1976jv Namely, the notion of application (interpreting forma% applieadon of A-terns) generalizes the classical Payhill-Shepherdson-Rogers definition of application in Pa, introduced to define Enlamesation Operaters (see Ro [9667!,

p.1431, Abstraction is defined accordingly,

An interesting fact is that these definitions do not depend on codings of pairs or of finite sets, wS:iZe tae elassica', ones do. Tiis doesn't affect the Rec~lssisn Theory one should be able :3 i;?mrc QU; on It, but does affect the model theory 0% A-C~ICMIUS

(see BB [1979]). NI~scever, fbr vz.,rict,s Teasons which should become clear in the next sections, Plotkin- Engeier3s structt~res are '%handy ppa~-tiic~da.slyin Engeler9s version: it is easy to grasp Ide ,:l'cuEtnerrs or which d'ie definitions and abstraction rely and to modify them for the p~rp3seaof he model theory of A-CBPGU~USwe aim at,

number of A-abstsactis-ns, Part JBI discusses the consequences in Combinatmy Algebras of an e~rly remark 3f SNadsworth (and Scott) on how to interpret the vvloss of jnf~flnation" which is ixpiQisrt in perhming eornb4watary reductions, as in any effective pmcess.

Section 2, ;oflowing Barendregt9s teminolsgy (Bar [1981]), deals with the local analysis of E~rgeler's models, i.e. sy nlacticnlly characterizes the true equalities in these models, Aetei,n,!!y the partid oder on these structures (i.e. set theoretic incIusion) matches perfectly weil *,he very n2t~11-dsyntactical partial order over A-terns, given by inelusion of Bsehm- trees +(the proofs are in Appendix W). This provides an algebraic characterization sf h-

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Section 3 gives a ssernantical c'har~cterization of A-terms of my 'inrite (and infinite) order, Le., for nEw, l~haracterizes the class of terms such as d\xl...x,.N according to...